Now can apply the lower-tail bound Hoeffding's inequality ([1]}, Theorem 1.10.12) to get
\(\begin{split}\textnormal {Pr}\left(\sum _{i=1}^{\alpha \beta } {\mathbb {1}_{i}} \le - c\sqrt{\beta \log \frac{1}{\delta }} + \mathbf {E}\sum _{i=1}^{\alpha \beta }{\mathbb {1}_{i}}\right) &\le \exp \left(-\frac{c^2\log \frac{1}{\delta }}{3}\right) \\&\le \delta /2\end{split}\) 
